Have you considered basing your project around a cheap bay-trail tablet?
Until fairly recently the bay-trail platform was plagued by processor bugs (which Intel didn't show much interest in fixing), and poor Linux support. There have also been Windows driver issues which means that most of them came shipped with a 32bit version of Windows 10 and a 32bit bootloader, even though the processor supports 64 bit. Because of this, they can be bought really cheaply at the moment.
However, the good news is that almost all of the the linux issues have now been fixed by a guy called Ian Morrison (Linuxium). I'm running Ubuntu on my baytrail tablet with his patches applied, and everything works great apart from the accelerometer, which is hardly a deal-breaker. I'm seriously considering putting Retropie on it, and using it as a poor man's switch.